Problem: Factor completely. $25x^2-30x+9=$
Solution: Both $25x^2$ and $9$ are perfect squares, since $25x^2=({5x})^2$ and $9=({3})^2$. Additionally, $30x$ is twice the product of the roots of $25x^2$ and $9$, since $30x=2({5x})( 3)$. $25x^2-30x+9 = ({5x})^2-2({5x})( 3)+({3})^2$ So we can use the square of a difference pattern to factor: ${a}^2 -2( a)( b)+ {b}^2 =({a}-{b})^2$ In this case, ${a}={5x}$ and ${b}={3}$ : $ ({5x})^2-2({5x})( 3)+({3})^2 =({5x}-{3})^2$ In conclusion, $25x^2-30x+9=(5x-3)^2$ Remember that you can always check your factorization by expanding it.